Timeline for Uniqueness of distributional solutions to the Poisson equation
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| S Apr 1, 2021 at 23:18 | history | edited | sharpend | CC BY-SA 4.0 |
added comment on tempered dists
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| S Apr 1, 2021 at 23:18 | history | suggested | Maximilian Janisch | CC BY-SA 4.0 |
added Weyl's Lemma
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| Apr 1, 2021 at 23:01 | review | Suggested edits | |||
| S Apr 1, 2021 at 23:18 | |||||
| Apr 1, 2021 at 18:11 | comment | added | Maximilian Janisch | @AlexandreEremenko Yes absolutely, but $A-B$ can be of the type $A-B: S(\mathbb R^n)\to\mathbb R, \phi\mapsto\int_{\mathbb R^n} p\phi\,\mathrm dx$ for a polynomial $p$. I am pretty sure that that is what Dallas meant 🙂 | |
| Apr 1, 2021 at 17:18 | comment | added | Alexandre Eremenko | There is no polynomial, except 0, in the space $S(R^n)$, if I correctly understand that $S(R^n)$ is the space of Schwartz test functions. | |
| Apr 1, 2021 at 16:18 | vote | accept | Maximilian Janisch | ||
| Apr 1, 2021 at 16:18 | comment | added | Maximilian Janisch | @DanieleTampieri Ohhh right, since $A-B$ could be any weak solution to the Laplace equation. Took me a while to see 😆! Thanks to both of you! | |
| Apr 1, 2021 at 16:10 | comment | added | Daniele Tampieri | Yes @MaximilianJanisch, you cannot always conclude that $A=B$, and this is in general true for every linear partial differential operator: the non homogeneous equation has a solution which is unique modulo a solution of the homogeneous equation. To make sure that this is only the null one, you must assume additional requirements as sharpened has exemplified. | |
| Apr 1, 2021 at 16:04 | comment | added | sharpend | Right, it just isn't true at that level of generality. ($A$ and $B$ could be different constants.) But at least you know what their difference must be. And thank you :) | |
| Apr 1, 2021 at 16:00 | comment | added | Maximilian Janisch | So for general $A,B$ we can't always conclude $A=B$ ? 😞 PS: I hope your French is doing well 🙂. | |
| Apr 1, 2021 at 15:57 | history | answered | sharpend | CC BY-SA 4.0 |